Divide
Rational Numbers
Division helps us find ONE GROUP, the NUMBER OF GROUPS, or a UNIT RATE when signed values are involved.
Yesterday: multiplication scaled a signed change. Today: division helps us undo that scaling by asking “per 1 group?” or “how many groups?”
What are we learning?
We are learning to divide rational numbers fluently and use division to find unit rates, equal shares, and missing values.
Success Criteria
Sign first. Students should be able to predict the sign and name the reciprocal step before doing the arithmetic.
Retrieve what you already know
Sign of a quotient
−18 ÷ 6
Reciprocal
reciprocal of −3/4
Multiply
(−3/4)(−8/9)
Unit rate
A tank loses 8 gallons in 4 hours. What is the change in 1 hour?
Total change ÷ number of equal intervals gives the change in 1 interval. That is a unit-rate idea.
What does one equal share look like?
Construction company income
The company ends the year with −$5,154.45 in income. The company is owned by 21 partners with equal shares.
Estimate first
If the loss were about −$5,250, then each share would be about −$250 because 21 × 250 ≈ 5,250.
Find the exact share
We need one equal share, so divide the total income by the number of partners.
Each partner’s portion of the income is −$245.45.
This asks for the value of one equal part, so division is the correct operation.
Division is connected to multiplication
Same situation, written another way
Division can be viewed as asking for the missing factor.
How much loss belongs to one partner?
−5,154.45 ÷ 21 = ?
Negative ÷ positive = negative.
Multiply when you know the unit amount and want the total. Divide when you know the total and want one equal part or the number of groups.
Fact-family thinking
Do not ignore the sign. Equal shares of a loss are still negative.
What change happens in 1 minute?
Estimate first
If −3.6 gallons drains in 6 minutes, then each minute should be about −0.6 gallon.
Choose the operation
We know the total change and want the amount in 1 equal minute.
Estimate → write division → find one equal part.
Rewrite division. Then use multiplication.
Each of the 6 equal parts is −3/5 gallon.
(−18/5) ÷ 6 = (−18/5)(1/6)
(−18/5)(1/6) = −18/30 = −3/5
The change after 1 minute is −3/5 gallon.
Division of rational numbers uses ideas you already know: integer sign rules and multiplication by a reciprocal.
Only the divisor becomes a reciprocal. The dividend stays the same.
Division becomes multiplication by the reciprocal
Vocabulary
the number being divided
the number you divide by
the answer to a division problem
two numbers whose product is 1
The rewrite rule
More precisely, divide by a number means multiply by its reciprocal.
Reciprocal examples
You cannot divide by 0 because 0 has no reciprocal.
What sign will the quotient have?
If the signs of the dividend and divisor are the same, the quotient is positive. If the signs are different, the quotient is negative.
Can you identify the reciprocal quickly?
A reciprocal flips the fraction. Whole numbers can be written over 1 first.
Rewrite the quotient as multiplication
A complex fraction has a fraction in the numerator, the denominator, or both.
Positive ÷ negative = negative.
11/3 × (−3/2)
−33/6
−11/2 = −5 1/2
Dividing by −2/3 is the same as multiplying by its reciprocal, −3/2.
Ask these two questions every time
Use the signs of the dividend and divisor.
Only the divisor flips.
Do not flip both fractions. Keep the dividend. Flip only the divisor.
Predict the sign. Then choose the arithmetic method.
A: class solves together. B–C: students explain. D: students lead.
How long does the diving bird take to reach the sea bottom?
Known information
Rate of change: −0.06 km/min
Target location: −3/4 km
Question
How many minutes does it take for the bird’s location to change by −0.06 km each minute until it reaches −3/4 km?
(−3/4) ÷ (−0.06)
Negative ÷ negative = positive.
−0.75 ÷ (−0.06) = 12.5
It takes 12.5 minutes for the diving bird to reach the sea bottom.
Should this situation use multiplication or division?
Meteorologist forecast
The temperature is currently −15°C. It will decrease by 4/9°C each hour. What will the temperature be in 1.5 hours?
Model the change
(−4/9)(1.5)
We know the rate per hour and the number of hours, so we multiply to find the total change.
Compute
(−4/9)(3/2) = −2/3
Now add the change to the current temperature:
−15 + (−2/3) = −15 2/3°C
Use division when you need a unit rate, one share, or the number of groups. Use multiplication when you already know the rate and time (or number of groups) and want the total change.
Can you separate sign reasoning from the arithmetic?
A. Predict the sign
B. What kind of situation or form?
C. Solve
If a student misses the sign but gets the magnitude, reteach the sign rule — not the reciprocal step.
Quick self-check before the exit ticket
1. Explain the sign
If the dividend is negative and the divisor is negative, what sign will the quotient have?
2. Complex fraction idea
How is a complex fraction related to division?
3. Which operation?
You know a total loss and the number of equal days. Do you multiply or divide to find the loss per day?
4. Example quotient
(3/8) ÷ (−2/9)
Show what you know
Decimal quotient
10.584 ÷ (−0.42)
Find the quotient and explain why the sign is correct.
Reciprocal rewrite
(−5/6) ÷ (1/8)
Rewrite the division as multiplication, then solve.
Unit rate
A tank loses 1 3/5 mL in 10 minutes.
What is the change in the volume in 1 minute?
Independent practice for Lesson 1.5
STAAR-style: choose the equivalent multiplication expression.
#17 is a unit-rate problem. Students should represent the change as a negative value because the tank is leaking.